Rosetta Code

Ternary logic

Revision as of 20:17, 8 September 2011 by rosettacode>TobyK (→‎Using a general binary -> ternary transform)

In logic, a three-valued logic (also trivalent, ternary, or trinary logic, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Łukasiewicz, Lewis and Sulski. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.

Task
Ternary logic
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Ternary logic. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)
Example Ternary Logic Operators in Truth Tables:
not a
¬
True False
Maybe Maybe
False True
a and b
∧ True Maybe False
True True Maybe False
Maybe Maybe Maybe False
False False False False
a or b
∨ True Maybe False
True True True True
Maybe True Maybe Maybe
False True Maybe False
if a then b
⊃ True Maybe False
True True Maybe False
Maybe True Maybe Maybe
False True True True
a is equivalent to b
≡ True Maybe False
True True Maybe False
Maybe Maybe Maybe Maybe
False False Maybe True

Task:

  • Define a new type that emulates ternary logic by storing data trits.
  • Given all the binary logic operators of the original programming language, reimplement these operators for the new Ternary logic type trit.
  • Generate a sampling of results using trit variables.
  • Kudos for actually thinking up a test case algorithm where ternary logic is intrinsically useful, optimises the test case algorithm and is preferable to binary logic.

Note: Setun (Сетунь) was a balanced ternary computer developed in 1958 at Moscow State University. The device was built under the lead of Sergei Sobolev and Nikolay Brusentsov. It was the only modern ternary computer, using three-valued ternary logic

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.

File: Ternary_logic.a68 <lang algol68># -*- coding: utf-8 -*- #

MODE TRIT = STRUCT(BITS trit); INT trit width = 1, trit base = 3; FORMAT trit fmt = $c("⌊","⌈","?" #|"~"#)$;

  1. These values treated are as per "Balanced ternary" #
  2. eg true=1, maybe=0, false=-1 #

TRIT true =INITTRIT 4r1 #⌈#, maybe=INITTRIT 4r0 #?#, 

    false=INITTRIT 4r2 #⌊#;

TRIT flip=true, flop=false, flap=maybe;

OP REPR = (TRIT t)STRING: 

 IF   t = false THEN "⌊"
 ELIF t = maybe THEN "?"
 ELIF t = true  THEN "⌈"
 ELSE raise value error(("invalid TRIT value",INITINT t));~
 FI;
  2. Define some OPerators for coercing MODES #

OP INITTRIT = (BOOL in)TRIT: 

 (in|true|false);

OP B = (TRIT in)BOOL: 

 (in=true|TRUE|:in=false|FALSE|
   raise value error(("invalid TRIT to BOOL coercion: """, REPR in,""""));~
 );
  1. These values treated are as per "Balanced ternary" #
  2. n.b true=1, maybe=0, false=-1 #
  3. Warning: BOOL ABS FALSE (0) is not the same as TRIT ABS false (-1) #

OP INITINT = (TRIT t)INT: 

 IF   t=true  THEN 1
 ELIF t=maybe THEN 0
 ELIF t=false THEN -1
 ELSE raise value error(("invalid TRIT value",REPR t));~
 FI;

OP INITTRIT = (INT in)TRIT: ( 

 TRIT out;
 trit OF out:= trit OF
   IF   in= 1 THEN true
   ELIF in= 0 THEN maybe
   ELIF in=-1 THEN false
   ELSE raise value error(("invalid TRIT value",in));~
   FI;
 out

);

OP INITTRIT = (BITS b)TRIT: 

 (TRIT out; trit OF out:=b; out);
  2. Define the OPerators for the TRIT MODE #
  1. These can be optimised by peekng at the binary value #
  2. These operators are as per "Balanced ternary" #
  3. Warning: "both" is ignored as it isn't Ternary #

OP LT = (TRIT a,b)BOOL: a EQ false AND b NE false OR a EQ maybe AND b EQ true, 

  LE = (TRIT a,b)BOOL: a EQ b OR a LT b,
  EQ = (TRIT a,b)BOOL: trit OF a = trit OF b,
  NE = (TRIT a,b)BOOL: NOT (a EQ b),
  GE = (TRIT a,b)BOOL: NOT (a LT b),
  GT = (TRIT a,b)BOOL: NOT (a LE b);
  1. A solo, unique and rather confusing CMP OPerator #

PRIO CMP = 5; OP CMP = (TRIT a,b)TRIT: 

 IF a < b THEN false
 ELIF a = b THEN maybe
 ELIF a > b THEN true
 FI;
  1. ASCII OPerators #

OP < = (TRIT a,b)BOOL: a LT b, 

  <= = (TRIT a,b)BOOL: a LE b,
   = = (TRIT a,b)BOOL: a EQ b,
  /= = (TRIT a,b)BOOL: a NE b,
  >= = (TRIT a,b)BOOL: a GE b,
  >  = (TRIT a,b)BOOL: a GT b;
  1. Non ASCII OPerators

OP ≤ = (TRIT a,b)BOOL: a LE b, 

  ≠ = (TRIT a,b)BOOL: a NE b,
  ≥ = (TRIT a,b)BOOL: a GE b;

OP - = (TRIT t)TRIT: 

 IF   t=maybe THEN maybe
 ELIF t=true THEN false
 ELIF t=false THEN true
 ELSE raise value error(("invalid TRIT value",REPR t)); ~
 FI;
  1. Warning: This routine ASSIGNS "out" AND returns "carry" #

OP +:= = (REF TRIT out, TRIT arg)TRIT: 

 IF   out = maybe THEN out :=  arg; maybe
 ELIF arg = maybe THEN # out:= out# arg
 ELIF out =  arg THEN out := -out; arg
 ELIF out = -arg THEN out:=maybe; maybe
 ELSE raise value error((REPR out," + ",REPR arg)); ~
 FI;

OP + = (TRIT a, b)TRIT: 

 (TRIT out:=a; VOID(out+:=b); out);

OP - = (TRIT a, b)TRIT: 

 a + -b;

OP * = (TRIT a, b)TRIT: 

 IF   a = maybe OR b = maybe THEN maybe
 ELIF a = b THEN true
 ELSE false
 FI;

OP ODD = (TRIT t)BOOL: 

 t /= maybe;

COMMENT 

 Kleene logic truth tables:

END COMMENT

OP AND = (TRIT a,b)TRIT: ( 

 [,]TRIT(
   # ∧      maybe, true,  false,  #
   #maybe# (maybe, maybe, false),
   #true#  (maybe, true,  false),
   #false# (false, false, false)
 )[@0,@0][ABS trit OF a, ABS trit OF b]

);

OP OR = (TRIT a,b)TRIT: ( 

 [,]TRIT(
   # ∨      maybe, true, false,  #
   #maybe# (maybe, true, maybe),
   #true#  (true,  true, true),
   #false# (maybe, true, false)
 )[@0,@0][ABS trit OF a, ABS trit OF b]

);

PRIO IMPLIES = 1; # 1.9 # OP IMPLIES = (TRIT a,b)TRIT: ( 

 [,]TRIT(
   # ⊃      maybe, true, false, #
   #maybe# (maybe, true, maybe),
   #true#  (maybe, true, false),
   #false# (true,  true, true)
 )[@0,@0][ABS trit OF a, ABS trit OF b]

);

PRIO EQV = 1; # 1.8 # OP EQV = (TRIT a,b)TRIT: ( 

 [,]TRIT(
   # ≡      maybe, true,  false, #
   #maybe# (maybe, maybe, maybe),
   #true#  (maybe, true,  false),
   #false# (maybe, false, true)
 )[@0,@0][ABS trit OF a, ABS trit OF b]

);

  1. Non ASCII OPerators

OP ¬ = (TRIT a)TRIT: NOT b, 

  ∨ = (TRIT a,b)TRIT: a OR b,
  ∧ = (TRIT a,b)TRIT: a AND b,
  & = (TRIT a,b)TRIT: a AND b,
  ⊃ = (TRIT a,b)TRIT: a IMPLIES b,
  ≡ = (TRIT a,b)TRIT: a EQV b;
  1. </lang>File: test_Ternary_logic.a68

<lang algol68>#!/usr/local/bin/a68g --script #

  1. -*- coding: utf-8 -*- #

PR READ "prelude/general.a68" PR PR READ "Ternary_logic.a68" PR

[]TRIT trits = (false, maybe, true);

FORMAT col fmt = $" "g" "$; FORMAT row fmt = $l3(f(col fmt)"|")f(col fmt)$; FORMAT row sep fmt = $l3("---+")"---"l$;

PROC row sep = VOID: 

 printf(row sep fmt);

PROC title = (UTF op)VOID:( 

 print(("Operator: ",op));
 printf((row fmt," ",REPR false, REPR maybe, REPR true))

);


PROC print bool op table = (STRING op name, PROC(TRIT,TRIT)BOOL op)VOID: ( 

 title(op name);
 FOR i FROM LWB trits TO UPB trits DO
   row sep;
   TRIT ti = trits[i];
   printf((col fmt, REPR ti));
   FOR j FROM LWB trits TO UPB trits DO
     TRIT tj = trits[j];
     printf(($"|"$, col fmt, op(ti,tj)))
   OD
 OD;
 print(new line)

);

PROC print trit op table = (STRING op name, PROC(TRIT,TRIT)TRIT op)VOID: ( 

 title(op name);
 FOR i FROM LWB trits TO UPB trits DO
   row sep;
   TRIT ti = trits[i];
   printf((col fmt, REPR ti));
   FOR j FROM LWB trits TO UPB trits DO
     TRIT tj = trits[j];
     printf(($"|"$, col fmt, REPR op(ti,tj)))
   OD
 OD;
 print(new line)

);

printf(( 

 $"Comparitive table of coercions:"l$,
 $"  TRIT BOOL         INT"l$

));

FOR it FROM LWB trits TO UPB trits DO 

 TRIT t = trits[it];
 IF t = maybe THEN
   printf(($"  "g"  "$, REPR t, " ", INITINT t, $l$))
 ELSE
   printf(($"  "g"  "$, REPR t, B    t, INITINT t, $l$))
 FI

OD;

printf(( 

 $l"Specific test of the IMPLIES operator:"l$,
 $"  "g" implies "g" is "b("not ","")"a contradiction!"l$,
   B false,    B false,    B(false IMPLIES false),
   B false,    B true,     B(false IMPLIES true),
   B false,    REPR maybe, B(false IMPLIES maybe),
   B true,     B false,    B(true  IMPLIES false),
   B true,     B true,     B(true  IMPLIES true),
   REPR maybe, Btrue,      B(maybe IMPLIES true),
 $"  "g" implies "g" is "g" a contradiction!"l$,
   B true,     REPR maybe, REPR (true  IMPLIES maybe),
   REPR maybe, B false,    REPR (maybe IMPLIES false),
   REPR maybe, REPR maybe, REPR (maybe IMPLIES maybe),
 $l$

));

printf($l"Kleene logic truth table samples:"l$);

print trit op table("CMP", (TRIT a,b)TRIT: a CMP b); print trit op table("EQV", (TRIT a,b)TRIT: a EQV b); print trit op table("IMPLIES", (TRIT a,b)TRIT: a IMPLIES b); print trit op table("AND", (TRIT a,b)TRIT: a AND b); print trit op table("OR", (TRIT a,b)TRIT: a OR b) CO; print trit op table("+", (TRIT a,b)TRIT: a + b); print trit op table("-", (TRIT a,b)TRIT: a - b); print trit op table("*", (TRIT a,b)TRIT: a * b); print bool op table("EQ", (TRIT a,b)BOOL: a EQ b); print bool op table("<=", (TRIT a,b)BOOL: a <= b) END CO</lang> Output: 

Comparitive table of coercions:
  TRIT BOOL         INT
  ⌊    F             -1  
  ?                  +0  
  ⌈    T             +1  

Specific test of the IMPLIES operator:
  F implies F is not a contradiction!
  F implies T is not a contradiction!
  F implies ? is not a contradiction!
  T implies F is a contradiction!
  T implies T is not a contradiction!
  ? implies T is not a contradiction!
  T implies ? is ? a contradiction!
  ? implies F is ? a contradiction!
  ? implies ? is ? a contradiction!


Kleene logic truth table samples:
Operator: CMP
   | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ? | ⌊ | ⌊ 
---+---+---+---
 ? | ⌈ | ? | ⌊ 
---+---+---+---
 ⌈ | ⌈ | ⌈ | ? 
Operator: EQV
   | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌈ | ? | ⌊ 
---+---+---+---
 ? | ? | ? | ? 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 
Operator: IMPLIES
   | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌈ | ⌈ | ⌈ 
---+---+---+---
 ? | ? | ? | ⌈ 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 
Operator: AND
   | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌊ | ⌊ | ⌊ 
---+---+---+---
 ? | ⌊ | ? | ? 
---+---+---+---
 ⌈ | ⌊ | ? | ⌈ 
Operator: OR
   | ⌊ | ? | ⌈ 
---+---+---+---
 ⌊ | ⌊ | ? | ⌈ 
---+---+---+---
 ? | ? | ? | ⌈ 
---+---+---+---
 ⌈ | ⌈ | ⌈ | ⌈

C

Implementing logic using lookup tables

<lang c>#include <stdio.h>

typedef enum { 

 TRITTRUE,  /* In this enum, equivalent to integer value 0 */
 TRITMAYBE, /* In this enum, equivalent to integer value 1 */
 TRITFALSE  /* In this enum, equivalent to integer value 2 */

} trit;

/* We can trivially find the result of the operation by passing 

  the trinary values as indeces into the lookup tables' arrays. */

trit tritNot[3] = {TRITFALSE , TRITMAYBE, TRITTRUE}; trit tritAnd[3][3] = { {TRITTRUE, TRITMAYBE, TRITFALSE}, 

                      {TRITMAYBE, TRITMAYBE, TRITFALSE},
                      {TRITFALSE, TRITFALSE, TRITFALSE} };

trit tritOr[3][3] = { {TRITTRUE, TRITTRUE, TRITTRUE}, 

                     {TRITTRUE, TRITMAYBE, TRITMAYBE},
                     {TRITTRUE, TRITMAYBE, TRITFALSE} };

trit tritThen[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE}, 

                       { TRITTRUE, TRITMAYBE, TRITMAYBE},
                       { TRITTRUE, TRITTRUE, TRITTRUE } };

trit tritEquiv[3][3] = { { TRITTRUE, TRITMAYBE, TRITFALSE}, 

                        { TRITMAYBE, TRITMAYBE, TRITMAYBE},
                        { TRITFALSE, TRITMAYBE, TRITTRUE } };

/* Everything beyond here is just demonstration */

const char* tritString[3] = {"T", "?", "F"};

void demo_binary_op(trit operator[3][3], char* name) { 

 trit operand1 = TRITTRUE; /* Declare. Initialize for CYA */
 trit operand2 = TRITTRUE; /* Declare. Initialize for CYA */

 /* Blank line */
 printf("\n");

 /* Demo this operator */
 for( operand1 = TRITTRUE; operand1 <= TRITFALSE; ++operand1 )
 {
   for( operand2 = TRITTRUE; operand2 <= TRITFALSE; ++operand2 )
   {
     printf("%s %s %s: %s\n", tritString[operand1],
                              name,
                              tritString[operand2],
                              tritString[operator[operand1][operand2]]);
   }
 }

}

int main() { 

 trit op1 = TRITTRUE; /* Declare. Initialize for CYA */
 trit op2 = TRITTRUE; /* Declare. Initialize for CYA */

 /* Demo 'not' */
 for( op1 = TRITTRUE; op1 <= TRITFALSE; ++op1 )
 {
   printf("Not %s: %s\n", tritString[op1], tritString[tritNot[op1]]);
 }
 demo_binary_op(tritAnd, "And");
 demo_binary_op(tritOr, "Or");
 demo_binary_op(tritThen, "Then");
 demo_binary_op(tritEquiv, "Equiv");



 return 0;

}</lang>

Output: <lang text>Not T: F Not ?: ? Not F: T

T And T: T T And ?: ? T And F: F ? And T: ? ? And ?: ? ? And F: F F And T: F F And ?: F F And F: F

T Or T: T T Or ?: T T Or F: T ? Or T: T ? Or ?: ? ? Or F: ? F Or T: T F Or ?: ? F Or F: F

T Then T: T T Then ?: ? T Then F: F ? Then T: T ? Then ?: ? ? Then F: ? F Then T: T F Then ?: T F Then F: T

T Equiv T: T T Equiv ?: ? T Equiv F: F ? Equiv T: ? ? Equiv ?: ? ? Equiv F: ? F Equiv T: F F Equiv ?: ? F Equiv F: T</lang>

Using functions

<lang c>#include <stdio.h>

typedef enum { t_F = -1, t_M, t_T } trit;

trit t_not (trit a) { return -a; } trit t_and (trit a, trit b) { return a < b ? a : b; } trit t_or (trit a, trit b) { return a > b ? a : b; } trit t_eq (trit a, trit b) { return a * b; } trit t_imply(trit a, trit b) { return -a > b ? -a : b; } char t_s(trit a) { return "F?T"[a + 1]; }

  1. define forall(a) for(a = t_F; a <= t_T; a++)

void show_op(trit (*f)(trit, trit), char *name) { trit a, b; printf("\n[%s]\n F ? T\n -------", name); forall(a) { printf("\n%c |", t_s(a)); forall(b) printf(" %c", t_s(f(a, b))); } puts(""); }

int main(void) { trit a;

puts("[Not]"); forall(a) printf("%c | %c\n", t_s(a), t_s(t_not(a)));

show_op(t_and, "And"); show_op(t_or, "Or"); show_op(t_eq, "Equiv"); show_op(t_imply, "Imply");

return 0; }</lang>output<lang>[Not] F | T ? | ? T | F

[And] 

   F ? T
 -------

F | F F F ? | F ? ? T | F ? T

[Or] 

   F ? T
 -------

F | F ? T ? | ? ? T T | T T T

[Equiv] 

   F ? T
 -------

F | T ? F ? | ? ? ? T | F ? T

[Imply] 

   F ? T
 -------

F | T T T ? | ? ? T T | F ? T</lang>

Delphi

<lang delphi>unit TrinaryLogic;

interface

//Define our own type for ternary logic. //This is actually still a Boolean, but the compiler will use distinct RTTI information. type 

   TriBool = type Boolean;

const 

   TTrue:TriBool = True;
   TFalse:TriBool = False;
   TMaybe:TriBool = TriBool(2);

function TVL_not(Value: TriBool): TriBool; function TVL_and(A, B: TriBool): TriBool; function TVL_or(A, B: TriBool): TriBool; function TVL_xor(A, B: TriBool): TriBool; function TVL_eq(A, B: TriBool): TriBool;

implementation

Uses 

   SysUtils;

function TVL_not(Value: TriBool): TriBool; begin 

   if Value = True Then
       Result := TFalse
   else If Value = False Then
       Result := TTrue
   else
       Result := Value;

end;

function TVL_and(A, B: TriBool): TriBool; begin 

   Result := TriBool(Iff(Integer(A * B) > 1, Integer(TMaybe), A * B));

end;

function TVL_or(A, B: TriBool): TriBool; begin 

   Result := TVL_not(TVL_and(TVL_not(A), TVL_not(B)));

end;

function TVL_xor(A, B: TriBool): TriBool; begin 

   Result := TVL_and(TVL_or(A, B), TVL_not(TVL_or(A, B)));

end;

function TVL_eq(A, B: TriBool): TriBool; begin 

   Result := TVL_not(TVL_xor(A, B));

end;

end.</lang>

And that's the reason why you never on no account ever should compare against the values of True or False unless you intent ternary logic!

An alternative version would be using an enum type <lang delphi>type TriBool = (tbFalse, tbMaybe, tbTrue);</lang> and defining a set of constants implementing the above tables: <lang delphi>const 

   tvl_not: array[TriBool] = (tbTrue, tbMaybe, tbFalse);
   tvl_and: array[TriBool, TriBool] = (
       (tbFalse, tbFalse, tbFalse),
       (tbFalse, tbMaybe, tbMaybe),
       (tbFalse, tbMaybe, tbTrue),
       );
   tvl_or: array[TriBool, TriBool] = (
       (tbFalse, tbMaybe, tbTrue),
       (tbMaybe, tbMaybe, tbTrue),
       (tbTrue, tbTrue, tbTrue),
       );
   tvl_xor: array[TriBool, TriBool] = (
       (tbFalse, tbMaybe, tbTrue),
       (tbMaybe, tbMaybe, tbMaybe),
       (tbTrue, tbMaybe, tbFalse),
       );
   tvl_eq: array[TriBool, TriBool] = (
       (tbTrue, tbMaybe, tbFalse),
       (tbMaybe, tbMaybe, tbMaybe),
       (tbFalse, tbMaybe, tbTrue),
       );

</lang>

That's no real fun, but lookup can then be done with <lang delphi>Result := tvl_and[A, B];</lang>

Icon and Unicon

The following example works in both Icon and Unicon. There are a couple of comments on the code that pertain to the task requirements:

  • Strictly speaking there are no binary values in Icon and Unicon. There are a number of flow control operations that result in expression success (and a result) or failure which affects flow. As a result there really isn't a set of binary operators to map into ternary. The example provides the minimum required by the task plus xor.
  • The code below does not define a data type as it doesn't really make sense in this case. Icon and Unicon can create records which would be overkill and clumsy in this case. Unicon can create objects which would also be overkill. The only remaining option is to reinterpret one of the existing types as ternary values. The code below implements balanced ternary values as integers in order to simplify several of the functions.
  • The use of integers doesn't really support strings of trits well. While there is a function showtrit to ease display a converse function to decode character trits in a string is not included.


<lang Icon>$define TRUE 1 $define FALSE -1 $define UNKNOWN 0

invocable all link printf

procedure main() # demonstrate ternary logic

ufunc := ["not3"] bfunc := ["and3", "or3", "xor3", "eq3", "ifthen3"]

every f := !ufunc do { # display unary functions 

  printf("\nunary function=%s:\n",f)
  every t1 := (TRUE | FALSE | UNKNOWN) do
     printf(" %s : %s\n",showtrit(t1),showtrit(not3(t1)))
  }


every f := !bfunc do { # display binary functions 

  printf("\nbinary function=%s:\n     ",f)
  every t1 := (&null | TRUE | FALSE | UNKNOWN) do { 
     printf(" %s : ",showtrit(\t1))
     every t2 := (TRUE | FALSE | UNKNOWN | &null) do {
        if /t1 then printf("  %s",showtrit(\t2)|"\n")
        else printf("  %s",showtrit(f(t1,\t2))|"\n")
        }
     }
  }

end

procedure showtrit(a) #: return printable trit of error if invalid return case a of {TRUE:"T";FALSE:"F";UNKNOWN:"?";default:runerr(205,a)} end

procedure istrit(a) #: return value of trit or error if invalid return (TRUE|FALSE|UNKNOWN|runerr(205,a)) = a end

procedure not3(a) #: not of trit or error if invalid return FALSE * istrit(a) end

procedure and3(a,b) #: and of two trits or error if invalid return min(istrit(a),istrit(b)) end

procedure or3(a,b) #: or of two trits or error if invalid return max(istrit(a),istrit(b)) end

procedure eq3(a,b) #: equals of two trits or error if invalid return istrit(a) * istrit(b) end

procedure ifthen3(a,b) #: if trit then trit or error if invalid return case istrit(a) of { TRUE: istrit(b) ; UNKNOWN: or3(a,b); FALSE: TRUE } end

procedure xor3(a,b) #: xor of two trits or error if invalid return not3(eq3(a,b)) end</lang>

Library: Icon Programming Library

printf.icn provides support for the printf family of functions

Output:

unary function=not3:
 T : F
 F : T
 ? : ?

binary function=and3:
       T  F  ?
 T :   T  F  ?
 F :   F  F  F
 ? :   ?  F  ?

binary function=or3:
       T  F  ?
 T :   T  T  T
 F :   T  F  ?
 ? :   T  ?  ?

binary function=xor3:
       T  F  ?
 T :   F  T  ?
 F :   T  F  ?
 ? :   ?  ?  ?

binary function=eq3:
       T  F  ?
 T :   T  F  ?
 F :   F  T  ?
 ? :   ?  ?  ?

binary function=ifthen3:
       T  F  ?
 T :   T  F  ?
 F :   T  T  T
 ? :   T  ?  ?


J

The designers of J felt that user defined types were harmful, so that part of the task will not be supported here.

Instead:

true: 1 false: 0 maybe: 0.5

<lang j>not=: -. and=: <. or =: >. if =: (>. -.)"0~ eq =: (<.&-. >. <.)"0</lang>

Example use:

<lang j> not 0 0.5 1 1 0.5 0 

  0 0.5 1 and/ 0 0.5 1

0 0 0 0 0.5 0.5 0 0.5 1 

  0 0.5 1 or/ 0 0.5 1
 0 0.5 1

0.5 0.5 1 

 1   1 1

  0 0.5 1 if/ 0 0.5 1
 1   1 1

0.5 0.5 1 

 0 0.5 1

  0 0.5 1 eq/ 0 0.5 1
 1 0.5   0

0.5 0.5 0.5 

 0 0.5   1</lang>

Note that this implementation is a special case of "fuzzy logic" (using a limited set of values).

Note that while >. and <. could be used for boolean operations instead of J's +. and *., the identity elements for >. and <. are not boolean values, but are negative and positive infinity. See also: Boolean ring

Java

Works with: Java version 1.5+

<lang java5>public class Logic{ public static enum Trit{ TRUE, MAYBE, FALSE;

public Trit and(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return (other == FALSE) ? FALSE : MAYBE; }else{ return FALSE; } }

public Trit or(Trit other){ if(this == TRUE){ return TRUE; }else if(this == MAYBE){ return (other == TRUE) ? TRUE : MAYBE; }else{ return other; } }

public Trit tIf(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return (other == TRUE) ? TRUE : MAYBE; }else{ return TRUE; } }

public Trit not(){ if(this == TRUE){ return FALSE; }else if(this == MAYBE){ return MAYBE; }else{ return TRUE; } }

public Trit equals(Trit other){ if(this == TRUE){ return other; }else if(this == MAYBE){ return MAYBE; }else{ return other.not(); } } } public static void main(String[] args){ for(Trit a:Trit.values()){ System.out.println("not " + a + ": " + a.not()); } for(Trit a:Trit.values()){ for(Trit b:Trit.values()){ System.out.println(a+" and "+b+": "+a.and(b)+ "\t "+a+" or "+b+": "+a.or(b)+ "\t "+a+" implies "+b+": "+a.tIf(b)+ "\t "+a+" = "+b+": "+a.equals(b)); } } } }</lang> Output: 

not TRUE: FALSE
not MAYBE: MAYBE
not FALSE: TRUE
TRUE and TRUE: TRUE	 TRUE or TRUE: TRUE	 TRUE implies TRUE: TRUE	 TRUE = TRUE: TRUE
TRUE and MAYBE: MAYBE	 TRUE or MAYBE: TRUE	 TRUE implies MAYBE: MAYBE	 TRUE = MAYBE: MAYBE
TRUE and FALSE: FALSE	 TRUE or FALSE: TRUE	 TRUE implies FALSE: FALSE	 TRUE = FALSE: FALSE
MAYBE and TRUE: MAYBE	 MAYBE or TRUE: TRUE	 MAYBE implies TRUE: TRUE	 MAYBE = TRUE: MAYBE
MAYBE and MAYBE: MAYBE	 MAYBE or MAYBE: MAYBE	 MAYBE implies MAYBE: MAYBE	 MAYBE = MAYBE: MAYBE
MAYBE and FALSE: FALSE	 MAYBE or FALSE: MAYBE	 MAYBE implies FALSE: MAYBE	 MAYBE = FALSE: MAYBE
FALSE and TRUE: FALSE	 FALSE or TRUE: TRUE	 FALSE implies TRUE: TRUE	 FALSE = TRUE: FALSE
FALSE and MAYBE: FALSE	 FALSE or MAYBE: MAYBE	 FALSE implies MAYBE: TRUE	 FALSE = MAYBE: MAYBE
FALSE and FALSE: FALSE	 FALSE or FALSE: FALSE	 FALSE implies FALSE: TRUE	 FALSE = FALSE: TRUE

OCaml

<lang ocaml>type trit = True | False | Maybe

let t_not = function 

 | True -> False
 | False -> True
 | Maybe -> Maybe

let t_and a b = 

 match a with
 | True -> b
 | False -> False
 | Maybe ->
     match b with
     | False -> False
     | _ -> Maybe

let t_or a b = 

 match a with
 | True -> True
 | False -> b
 | Maybe ->
     match b with
     | True -> True
     | _ -> Maybe

let t_eq a b = 

 match a with
 | True -> b
 | Maybe -> Maybe
 | False ->
     match b with
     | True -> False
     | False -> True
     | Maybe -> Maybe

let t_imply a b = 

 match a, b with
 | _, True -> True
 | True, b -> b
 | False, _ -> True
 | Maybe, _ -> Maybe

let string_of_trit = function 

 | True -> "True"
 | False -> "False"
 | Maybe -> "Maybe"

let () = 

 let values = [| True; Maybe; False |] in
 let f = string_of_trit in
 Array.iter (fun v -> Printf.printf "Not %s: %s\n" (f v) (f (t_not v))) values;
 print_newline ();
 let print op str =
   Array.iter (fun a ->
     Array.iter (fun b ->
       Printf.printf "%s %s %s: %s\n" (f a) str (f b) (f (op a b))
     ) values
   ) values;
   print_newline ()
 in
 print t_and "And";
 print t_or "Or";
 print t_imply "Then";
 print t_eq "Equiv";
</lang>

Output: 

Not True: False
Not Maybe: Maybe
Not False: True

True And True: True
True And Maybe: Maybe
True And False: False
Maybe And True: Maybe
Maybe And Maybe: Maybe
Maybe And False: False
False And True: False
False And Maybe: False
False And False: False

True Or True: True
True Or Maybe: True
True Or False: True
Maybe Or True: True
Maybe Or Maybe: Maybe
Maybe Or False: Maybe
False Or True: True
False Or Maybe: Maybe
False Or False: False

True Then True: True
True Then Maybe: Maybe
True Then False: False
Maybe Then True: True
Maybe Then Maybe: Maybe
Maybe Then False: Maybe
False Then True: True
False Then Maybe: True
False Then False: True

True Equiv True: True
True Equiv Maybe: Maybe
True Equiv False: False
Maybe Equiv True: Maybe
Maybe Equiv Maybe: Maybe
Maybe Equiv False: Maybe
False Equiv True: False
False Equiv Maybe: Maybe
False Equiv False: True

Using a general binary -> ternary transform

Instead of writing all of the truth-tables by hand (and possibly introducing typos), we can construct a general binary -> ternary transform and apply it to any logical function we want: <lang OCaml>type trit = True | False | Maybe

let to_bin = function True -> [true] | False -> [false] | Maybe -> [true;false] let eval f x = 

  List.fold_left (fun l c -> List.fold_left (fun m d -> ((d c) :: m)) l f) [] x

let rec from_bin = 

  function [true] -> True | [false] -> False
  | h :: t -> (match (h, from_bin t) with
     (true,True) -> True | (false,False) -> False | _ -> Maybe)
  | _ -> Maybe

let to_ternary1 uop = fun x -> from_bin (eval [uop] (to_bin x)) let to_ternary2 bop = fun x y -> from_bin (eval (eval [bop] (to_bin x)) (to_bin y))

let t_not = to_ternary1 (not) let t_and = to_ternary2 (&&) let t_or = to_ternary2 (||) let t_equiv = to_ternary2 (=) let t_imply = to_ternary2 (fun p q -> (not p) || q)

let str = function True -> "True " | False -> "False" | Maybe -> "Maybe" let iterv f = List.iter f [True; False; Maybe]

let table1 s u = 

  print_endline ("\n"^s^":");
  iterv (fun v -> print_endline ("  "^(str v)^" -> "^(str (u v))));;

let table2 s b = 

  print_endline ("\n"^s^":");
  iterv (fun u ->
     iterv (fun v ->
        print_endline ("  "^(str u)^" "^(str v)^" -> "^(str (b u v)))));;

table1 "not" t_not;; table2 "and" t_and;; table2 "or" t_or;; table2 "equiv" t_equiv;; table2 "implies" t_imply;;</lang> Output: 

not:
  True  -> False
  False -> True 
  Maybe -> Maybe

and:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> False
  False False -> False
  False Maybe -> False
  Maybe True  -> Maybe
  Maybe False -> False
  Maybe Maybe -> Maybe

or:
  True  True  -> True 
  True  False -> True 
  True  Maybe -> True 
  False True  -> True 
  False False -> False
  False Maybe -> Maybe
  Maybe True  -> True 
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

equiv:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> False
  False False -> True 
  False Maybe -> Maybe
  Maybe True  -> Maybe
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

implies:
  True  True  -> True 
  True  False -> False
  True  Maybe -> Maybe
  False True  -> True 
  False False -> True 
  False Maybe -> True 
  Maybe True  -> True 
  Maybe False -> Maybe
  Maybe Maybe -> Maybe

Perl 6

Works with: niecza

Implementation: <lang perl6>enum Trit <Foo Moo Too>;

sub prefix:<¬> (Trit $a) { Trit(1-($a-1)) }

sub infix:<∧> is equiv(&infix:<*>) (Trit $a, Trit $b) { $a min $b } sub infix:<∨> is equiv(&infix:<+>) (Trit $a, Trit $b) { $a max $b }

sub infix:<→> is equiv(&infix:<..>) (Trit $a, Trit $b) { ¬$a max $b } sub infix:<≡> is equiv(&infix:<eq>) (Trit $a, Trit $b) { Trit(1 + ($a-1) * ($b-1)) }</lang> The precedence of each operator is specified as equivalent to an existing operator. We've taken the liberty of using an arrow for implication, to avoid confusing it with ⊃, (U+2283 SUPERSET OF).

To test, we use this code: <lang perl6>say '¬'; say "Too {¬Too}"; say "Moo {¬Moo}"; say "Foo {¬Foo}";

sub tbl (&op,$name) { 

   say ;
   say "$name   Too Moo Foo";
   say "   ╔═══════════";
   say "Too║{op Too,Too} {op Too,Moo} {op Too,Foo}";
   say "Moo║{op Moo,Too} {op Moo,Moo} {op Moo,Foo}";
   say "Foo║{op Foo,Too} {op Foo,Moo} {op Foo,Foo}";

}

tbl(&infix:<∧>, '∧'); tbl(&infix:<∨>, '∨'); tbl(&infix:<→>, '→'); tbl(&infix:<≡>, '≡');

say ; say 'Precedence tests should all print "Too":'; say ~( 

   Foo ∧ Too ∨ Too ≡ Too,
   Foo ∧ (Too ∨ Too) ≡ Foo,
   Too ∨ Too ∧ Foo ≡ Too,
   (Too ∨ Too) ∧ Foo ≡ Foo,

   ¬Too ∧ Too ∨ Too ≡ Too,
   ¬Too ∧ (Too ∨ Too) ≡ ¬Too,
   Too ∨ Too ∧ ¬Too ≡ Too,
   (Too ∨ Too) ∧ ¬Too ≡ ¬Too,

   Foo ∧ Too ∨ Foo → Foo ≡ Too,
   Foo ∧ Too ∨ Too → Foo ≡ Foo,

);</lang> Output: 

¬
Too Foo
Moo Moo
Foo Too

∧   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Moo Moo Foo
Foo║Foo Foo Foo

∨   Too Moo Foo
   ╔═══════════
Too║Too Too Too
Moo║Too Moo Moo
Foo║Too Moo Foo

→   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Too Moo Moo
Foo║Too Too Too

≡   Too Moo Foo
   ╔═══════════
Too║Too Moo Foo
Moo║Moo Moo Moo
Foo║Foo Moo Too

Precedence tests should all print "Too":
Too Too Too Too Too Too Too Too Too Too

Python

In Python, the keywords 'and', 'not', and 'or' are coerced to always work as boolean operators. I have therefore overloaded the boolean bitwise operators &, |, ^ to provide the required functionality. <lang python>class Trit(int): 

   def __new__(cls, value):
       if value == 'TRUE':
           value = 1
       elif value == 'FALSE':
           value = 0
       elif value == 'MAYBE':
           value = -1
       return super(Trit, cls).__new__(cls, value // (abs(value) or 1)) 

   def __repr__(self):
       if self > 0:
           return 'TRUE'
       elif self == 0:
           return 'FALSE'
       return 'MAYBE'

   def __str__(self):
       return repr(self)

   def __bool__(self):
       if self > 0:
           return True
       elif self == 0:
           return False
       else:
           raise ValueError("invalid literal for bool(): '%s'" % self)

   def __or__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][1]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][1]
           except:
               return NotImplemented

   def __ror__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][1]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][1]
           except:
               return NotImplemented

   def __and__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][0]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][0]
           except:
               return NotImplemented

   def __rand__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][0]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][0]
           except:
               return NotImplemented

   def __xor__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][2]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][2]
           except:
               return NotImplemented

   def __rxor__(self, other):
       if isinstance(other, Trit):
           return _ttable[(self, other)][2]
       else:
           try:
               return _ttable[(self, Trit(bool(other)))][2]
           except:
               return NotImplemented

   def __invert__(self):
       return _ttable[self]
   
   def __getattr__(self, name):
       if name in ('_n', 'flip'):
           # So you can do x._n == x.flip; the inverse of x
           # In Python 'not' is strictly boolean so we can't write `not x`
           # Same applies to keywords 'and' and 'or'.
           return _ttable[self]
       else:
           raise AttributeError


TRUE, FALSE, MAYBE = Trit(1), Trit(0), Trit(-1)

_ttable = { 

   #    A: -> flip_A
        TRUE: FALSE,
       FALSE:  TRUE,
       MAYBE: MAYBE,
   #     (A, B): -> (A_and_B, A_or_B, A_xor_B)
       (MAYBE, MAYBE): (MAYBE, MAYBE, MAYBE),
       (MAYBE, FALSE): (FALSE, MAYBE, MAYBE),
       (MAYBE,  TRUE): (MAYBE,  TRUE, MAYBE),
       (FALSE, MAYBE): (FALSE, MAYBE, MAYBE),
       (FALSE, FALSE): (FALSE, FALSE, FALSE),
       (FALSE,  TRUE): (FALSE,  TRUE,  TRUE),
       ( TRUE, MAYBE): (MAYBE,  TRUE, MAYBE),
       ( TRUE, FALSE): (FALSE,  TRUE,  TRUE),
       ( TRUE,  TRUE): ( TRUE,  TRUE, FALSE),
   }


values = ('FALSE', 'TRUE ', 'MAYBE')

print("\nTrit logical inverse, '~'") for a in values: 

   expr = '~%s' % a
   print('  %s = %s' % (expr, eval(expr)))

for op, ophelp in (('&', 'and'), ('|', 'or'), ('^', 'exclusive-or')): 

   print("\nTrit logical %s, '%s'" % (ophelp, op))
   for a in values:
       for b in values:
           expr = '%s %s %s' % (a, op, b)
           print('  %s = %s' % (expr, eval(expr)))</lang>
Output
Trit logical inverse, '~'
  ~FALSE = TRUE
  ~TRUE  = FALSE
  ~MAYBE = MAYBE

Trit logical and, '&'
  FALSE & FALSE = FALSE
  FALSE & TRUE  = FALSE
  FALSE & MAYBE = FALSE
  TRUE  & FALSE = FALSE
  TRUE  & TRUE  = TRUE
  TRUE  & MAYBE = MAYBE
  MAYBE & FALSE = FALSE
  MAYBE & TRUE  = MAYBE
  MAYBE & MAYBE = MAYBE

Trit logical or, '|'
  FALSE | FALSE = FALSE
  FALSE | TRUE  = TRUE
  FALSE | MAYBE = MAYBE
  TRUE  | FALSE = TRUE
  TRUE  | TRUE  = TRUE
  TRUE  | MAYBE = TRUE
  MAYBE | FALSE = MAYBE
  MAYBE | TRUE  = TRUE
  MAYBE | MAYBE = MAYBE

Trit logical exclusive-or, '^'
  FALSE ^ FALSE = FALSE
  FALSE ^ TRUE  = TRUE
  FALSE ^ MAYBE = MAYBE
  TRUE  ^ FALSE = TRUE
  TRUE  ^ TRUE  = FALSE
  TRUE  ^ MAYBE = MAYBE
  MAYBE ^ FALSE = MAYBE
  MAYBE ^ TRUE  = MAYBE
  MAYBE ^ MAYBE = MAYBE
Extra doodling in the Python shell
>>> values = (TRUE, FALSE, MAYBE)
>>> for a in values:
	for b in values:
		assert (a & ~b) | (b & ~a) == a ^ b

		
>>>

Ruby

Ruby, like Smalltalk, has two boolean classes: TrueClass for true and FalseClass for false. We add a third class, MaybeClass for MAYBE, and define ternary logic for all three classes.

We keep !a, a & b and so on for binary logic. We add !a.trit, a.trit & b and so on for ternary logic. The code for !a.trit uses def !, which works with Ruby 1.9, but fails as a syntax error with Ruby 1.8.

Works with: Ruby version 1.9

<lang ruby># trit.rb - ternary logic

  1. http://rosettacode.org/wiki/Ternary_logic

require 'singleton'

  1. MAYBE, the only instance of MaybeClass, enables a system of ternary
  2. logic using TrueClass#trit, MaybeClass#trit and FalseClass#trit.
  4. !a.trit # ternary not
  5. a.trit & b # ternary and
  6. a.trit | b # ternary or
  7. a.trit ^ b # ternary exclusive or
  8. a.trit == b # ternary equal
  10. Though +true+ and +false+ are internal Ruby values, +MAYBE+ is not.
  11. Programs may want to assign +maybe = MAYBE+ in scopes that use
  12. ternary logic. Then programs can use +true+, +maybe+ and +false+.

class MaybeClass 

 include Singleton

 #  maybe.to_s  # => "maybe"
 def to_s; "maybe"; end

end

MAYBE = MaybeClass.instance

class TrueClass 

 TritMagic = Object.new
 class << TritMagic
   def index; 0; end
   def !; false; end
   def & other; other; end
   def | other; true; end
   def ^ other; [false, MAYBE, true][other.trit.index]; end
   def == other; other; end
 end

 # Performs ternary logic. See MaybeClass.
 #  !true.trit        # => false
 #  true.trit & obj   # => obj
 #  true.trit | obj   # => true
 #  true.trit ^ obj   # => false, maybe or true
 #  true.trit == obj  # => obj
 def trit; TritMagic; end

end

class MaybeClass 

 TritMagic = Object.new
 class << TritMagic
   def index; 1; end
   def !; MAYBE; end
   def & other; [MAYBE, MAYBE, false][other.trit.index]; end
   def | other; [true, MAYBE, MAYBE][other.trit.index]; end
   def ^ other; MAYBE; end
   def == other; MAYBE; end
 end

 # Performs ternary logic. See MaybeClass.
 #  !maybe.trit        # => maybe
 #  maybe.trit & obj   # => maybe or false
 #  maybe.trit | obj   # => true or maybe
 #  maybe.trit ^ obj   # => maybe
 #  maybe.trit == obj  # => maybe
 def trit; TritMagic; end

end

class FalseClass 

 TritMagic = Object.new
 class << TritMagic
   def index; 2; end
   def !; true; end
   def & other; false; end
   def | other; other; end
   def ^ other; other; end
   def == other; [false, MAYBE, true][other.trit.index]; end
 end

 # Performs ternary logic. See MaybeClass.
 #  !false.trit        # => true
 #  false.trit & obj   # => false
 #  false.trit | obj   # => obj
 #  false.trit ^ obj   # => obj
 #  false.trit == obj  # => false, maybe or true
 def trit; TritMagic; end

end</lang>

This IRB session shows ternary not, and, or, equal.

<lang ruby>$ irb irb(main):001:0> require './trit' => true irb(main):002:0> maybe = MAYBE => maybe irb(main):003:0> !true.trit => false irb(main):004:0> !maybe.trit => maybe irb(main):005:0> maybe.trit & false => false irb(main):006:0> maybe.trit | true => true irb(main):007:0> false.trit == true => false irb(main):008:0> false.trit == maybe => maybe</lang>

This program shows all 9 outcomes from a.trit ^ b.

<lang ruby>require 'trit' maybe = MAYBE

[true, maybe, false].each do |a| 

 [true, maybe, false].each do |b|
   printf "%5s ^ %5s => %5s\n", a, b, a.trit ^ b
 end

end</lang> 

$ ruby -I. trit-xor.rb
 true ^  true => false
 true ^ maybe => maybe
 true ^ false =>  true
maybe ^  true => maybe
maybe ^ maybe => maybe
maybe ^ false => maybe
false ^  true =>  true
false ^ maybe => maybe
false ^ false => false

Tcl

The simplest way of doing this is by constructing the operations as truth tables. The code below uses an abbreviated form of truth table. <lang tcl>package require Tcl 8.5 namespace eval ternary { 

   # Code generator
   proc maketable {name count values} {

set sep "" for {set i 0; set c 97} {$i<$count} {incr i;incr c} { set v [format "%c" $c] lappend args $v; append key $sep "$" $v set sep "," } foreach row [split $values \n] { if {[llength $row]>1} { lassign $row from to lappend table $from [list return $to] } } proc $name $args \ [list ckargs $args]\;[concat [list switch -glob --] $key [list $table]] namespace export $name 

   }
   # Helper command to check argument syntax
   proc ckargs argList {

foreach var $argList { upvar 1 $var v switch -exact -- $v { true - maybe - false { continue } default { return -level 2 -code error "bad ternary value \"$v\"" } } } 

   }

   # The "truth" tables; “*” means “anything”
   maketable not 1 {

true false maybe maybe false true 

   }
   maketable and 2 {

true,true true false,* false *,false false * maybe 

   }
   maketable or 2 {

true,* true *,true true false,false false * maybe 

   }
   maketable implies 2 {

false,* true *,true true true,false false * maybe 

   }
   maketable equiv 2 {

*,maybe maybe maybe,* maybe true,true true false,false true * false 

   }

}</lang> Demonstrating: <lang tcl>namespace import ternary::* puts "x /\\ y == x \\/ y" puts " x | y || result" puts "-------+-------++--------" foreach x {true maybe false} { 

   foreach y {true maybe false} {

set z [equiv [and $x $y] [or $x $y]] puts [format " %-5s | %-5s || %-5s" $x $y $z] 

   }

}</lang> Output: 

x /\ y == x \/ y
 x     | y     || result
-------+-------++--------
 true  | true  || true 
 true  | maybe || maybe
 true  | false || false
 maybe | true  || maybe
 maybe | maybe || maybe
 maybe | false || maybe
 false | true  || false
 false | maybe || maybe
 false | false || true
Retrieved from "https://rosettacode.org/wiki/Ternary_logic?oldid=199197"